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Theorem mexval 35507
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mexval.k 𝐾 = (mTC‘𝑇)
mexval.e 𝐸 = (mEx‘𝑇)
mexval.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mexval 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mexval.e . 2 𝐸 = (mEx‘𝑇)
2 fveq2 6906 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3 mexval.k . . . . . 6 𝐾 = (mTC‘𝑇)
42, 3eqtr4di 2795 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5 fveq2 6906 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6 mexval.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6eqtr4di 2795 . . . . 5 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
84, 7xpeq12d 5716 . . . 4 (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9 df-mex 35492 . . . 4 mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10 fvex 6919 . . . . 5 (mTC‘𝑡) ∈ V
11 fvex 6919 . . . . 5 (mREx‘𝑡) ∈ V
1210, 11xpex 7773 . . . 4 ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
138, 9, 12fvmpt3i 7021 . . 3 (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14 xp0 6178 . . . . 5 (𝐾 × ∅) = ∅
1514eqcomi 2746 . . . 4 ∅ = (𝐾 × ∅)
16 fvprc 6898 . . . 4 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17 fvprc 6898 . . . . . 6 𝑇 ∈ V → (mREx‘𝑇) = ∅)
186, 17eqtrid 2789 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1918xpeq2d 5715 . . . 4 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
2015, 16, 193eqtr4a 2803 . . 3 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
2113, 20pm2.61i 182 . 2 (mEx‘𝑇) = (𝐾 × 𝑅)
221, 21eqtri 2765 1 𝐸 = (𝐾 × 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333   × cxp 5683  cfv 6561  mTCcmtc 35469  mRExcmrex 35471  mExcmex 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mex 35492
This theorem is referenced by:  mexval2  35508  msubff  35535  msubco  35536  msubff1  35561  mvhf  35563  msubvrs  35565
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